Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33759 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,16,19)(17,20,23,27)(18,26,24,22)(21,29,28,25);(6,8)(7,10)(9,13)(11,12)(15,19)(20,27)(21,28)(22,26);(8,13)(10,12)(15,20)(17,25)(18,24)(19,27)(21,22)(23,29)(26,28);(7,11)(10,12)(15,19)(20,27)(21,28)(22,26)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26747);